Determinants are not just a type of form (though differential forms and determinants have clear, obvious, and important similarities from the antisymmetry to the manner in which forms allow for integration over oriented manifolds) nor are determinants generally speaking tensors or tensor fields.
The determinant map for a vector space of dimension n is a completely anti-symmetric tensor with n entries. As such, it is a form (not a differential form) and a tensor (and not a tensor field).
In point of fact, even the mathematician's language gets confusing here as e.g., not all Cartesian tensors are in fact tensors (I forget the source, but in some work on applied mathematics or other the author introduced novel terminology for Cartesian tensors for which he apologized but pointed out that the reason was due to the fact that talking about tensors in general with Cartesian tensors specifically creates a misleading conception due to the fact that the logic between the two is the opposite that it seems; as I recall, the author made the comparison that talking about Cartesian tensors in relation to tensors in general was like asserting that "all trees are maple trees, but not all maple trees are trees). Indeed, even referring to an object as a tensor is often a shorthand conflating two (or more!) conceptually and formally different things, as e.g., the components of tensors vs. the abstract element of a set.
Not what I am talking about.
Yes. And the formal definition of a vector is that it is an element of a vector space. This is not a particularly helpful definition, even if one is given the necessary axioms to define a vector space, at least at a first pass.
It is actually greatly simplifying, especially if you vector space is infinite dimensional, like a Hilbert space or a function space.
Likewise, even the more formal definitions for tensors often involved defining tensors in terms of tensor product spaces or some similar "defining X as something that is X-like." As it is more useful in the case of tensors to immediately question whether an object indeed transforms like a tensor, not to mention more useful to build up the notion in terms of known structures such as vectors and forms, I prefer both (at least from an introductory perspective) and acknowledge that both physicists and mathematicians differ with respect to how they think it best to define such objects (I got into a lengthy argument/debate once with a mathematician over his conflation of points in R^n and vectors arguing that e.g., it makes no sense to speak of adding two points as this is a relation/operation one applies to vectors not points, and his general response was that such distinctions were unimportant in most contexts).
To focus on the components is to miss the trees for the forest. The components depend on the basis used. The transformations you mention simply describe how those components change when the basis is changed. In cases where there is no natural basis, the description as a vector space is MUCH more natural. In fact, it is usually best to get away from components and focus on the vectors or tensors themselves. Then, choose a basis (or not) for computations.
Many-body QM and relativistic QM deal with Fock spaces, true, but in general quantum systems even in the case of single-particle QM cannot be defined uniquely in terms of vectors but correspond more generally to rays due to the difficulties with the phase. But this is a good example for another reason: in both physics and mathematics it is common to refer to bases as you did without specifying whether one means e.g., a Hamel or Schauder basis. In finite-dimensional vector spaces, there is no use for such a distinction nor is it necessary to make it. In quantum mechanics, where one is often dealing systems that require a description that may actually need both countable and uncountable bases (or that are more generally just infinite, and therefore the distinction between countable and uncountable becomes relevant) this distinction is absolutely fundamental and is too often glossed over in treatments in the literature.
Yet another reason to discuss vectors and tensors in a coordinate free manner. Then, when you choose your basis (of either sort), there is no confusion about the components. The vector doesn't transform; the components do.
Also, it is quite possible for an infinite dimensional space to not have a Schauder basis (even if it is a Banach space).
But you are already conflating or mixing up different objects that one often deals with at differing levels of abstraction and structures. Manifolds need not be equipped with a vector space structure and often aren't, and indeed charts in many cases describe topological manifolds that aren't differentiable and therefore in both cases one often can't refer to basis vectors without already introducing a level of structure (namely, that of a vector space) that will hold for tangent spaces but not for the charts or coordinate maps in general. All the while, a central component of the general manifold structure is being glossed here- the atlas (not to mention the bundles one requires to even refer to e.g., the local tangent bundles or TM more generally)
The charts are the elements of the atlas. The vector and tensors are elements of some vector bundle. The fields are sections of those vector bundles.
Given a chart (a coordinate chart; an element of the atlas), there are coordinate lines where only one coordinate changes. The tangent vectors to those lines are a natural basis at every point in the chart. Usually, a physicist works in one of the bases obtained in this way.
The transformation formulas physicists like come directly from how the *components* of vectors and tensors change when a chart is changed (and thereby the basis produced from the chart).
BUT, there are other vector fields that give bases at every point. And, occasionally, these bases are useful for calculation. For example, it is occasionally useful in relativity to have two of the basis vectors be null vectors. These will not come from any coordinate chart, but they can simplify things in some cases.
Where exactly in Wald's book are you referring to?
Section 3.4b. The use of tetrads (which are non-coordinate bases of the tangent space) leads to some absolutely horrid notational nightmares.
A tetrad is simply four orthonormal vectors are each point. No need for multiple subscripts and superscript. The components of vectors and tensors can easily be done in the tetrad basis: the transformations of coordinates are precisely the same as for any change of basis. BUT there are no derivatives because it isn't a change of coordinate chart.
